Beyond Eighth Grade:

Functional Mathematics for Life and Work

MDS-1241

National Center for Research in Vocational EducationGraduate School of Education, University of California at Berkeley
2030 Addison Street, Suite 500, Berkeley, CA 94720-1674

Supported by The Office of Vocational and Adult Education, U.S. Department of Education
May 1999

FUNDING INFORMATION

Project Title:

National Center for Research in Vocational Education

Grant Number:

V051A30003-98A/V051A30004-98A

Act under which Funds Administered:

Carl D. Perkins Vocational Education Act
P.L. 98-524

Source of Grant:

Office of Vocational and Adult Education
U.S. Department of
EducationWashington, DC 20202

Grantee:

The Regents of the University of California
c/o National Center for Research in Vocational Education
2030 Addison Street, Suite 500
Berkeley, CA 94720

Director:

David Stern

Percent of Total Grant Financed by Federal Money:

100%

Dollar Amount of Federal Funds for Grant:

$4,500,000

Disclaimer:

This publication was prepared pursuant to a grant with the Office of
Vocational and Adult Education, U.S. Department of Education. Grantees
undertaking such projects under government sponsorship are encouraged to
express freely their judgement in professional and technical matters.
Points of view or opinions do not, therefore, necessarily represent
official U.S. Department of Education position or policy.

Discrimination:

Title VI of the Civil Rights Act of 1964 states: "No person in the
United States shall, on the ground of race, color, or national origin, be
excluded from participation in, be denied the benefits of, or be subjected
to discrimination under any program or activity receiving federal financial
assistance." Title IX of the Education Amendments of 1972 states: "No
person in the United States shall, on the basis of sex, be excluded from
participation in, be denied the benefits of, or be subjected to
discrimination under any education program or activity receiving federal
financial assistance." Therefore, the National Center for Research in
Vocational Education project, like every program or activity receiving
financial assistance from the U.S. Department of Education, must be
operated in compliance with these laws.

Acknowledgments

Beginning
in 1996, several industry associations and coalitions published occupational
skill standards to document the entry-level expectations of modern
high-performance industries. At the same time, the Institute on Education and
the Economy (IEE) at Teachers College, Columbia University, organized a series
of meetings to stimulate discussion about integrating academic and industry
skill standards. The meetings and subsequent publications were undertaken by
IEE in its capacity as a site of the National Center for Research in Vocational
Education. One such meeting, held in November 1997 at the Arden Conference
Center in Harriman, New York, focused on mathematics. Beyond Eight Grade: Functional Mathematics for Life and Workis one outcome of that meeting, a summary of issues and an interpretation of ideas that blend desires of employers with the expectations of academics. We wish to thank Thomas Bailey, Director of IEE, for suggesting and supporting this work, as well as IEE staff members Eric Larsen, Donna Merritt, and Lisa Rothman for their help with editorial and publishing details.

Executive Summary

Mathematics
is the key to many of the most secure and financially rewarding careers in
every sector of the economy. The impact of computers and information technology
can be seen not just in engineering and science, but in such diverse areas as
manufacturing, agriculture, health care, and advertising. To be prepared for
careers in virtually any industry, and especially for changing careers during a
lifetime, secondary school students need to learn a substantial core of
mathematics. However, this core is not like either the abstract pre-engineering
mathematics of the academic curriculum or the restricted topics of the
discredited "vocational math." New approaches are needed to meet today's
challenges.

For
most of this century, schools have been organized to prepare some students for
college and others for work. In the future, most jobs and all careers will
require some form of postsecondary education. Thus, all students, regardless of
career goals, can benefit from a mathematics curriculum that prepares them for
both work and higher education. A three-year high school program based on
"functional" mathematics can do just that: provide a strong background for
students entering the workforce as well as for those moving directly into
postsecondary education.

A
high school curriculum that helps all students master functional mathematics
would effectively address issues of both equity and competitiveness. All
students would have equal opportunity to master the mathematics required for
the new world of work. Moreover, a three-year core of functional mathematics
would give students a strong foundation from which to enter the workplace or
pursue advanced education.

Most
of the elements of functional mathematics are unsurprising. Percentages and
ratios; linear and quadratic equations; areas, angles, and volumes; and
exponential growth and trigonometric relations are included in any strong high
school mathematics program. Although some topics in functional mathematics are
uncommon (e.g., index numbers, tolerances, three-dimensional geometry, indirect
measurement, financial mathematics), most are taught in any high-quality high
school mathematics program. To achieve its goal of preparing students both for
work and for further education, functional mathematics respects the many parts
of the traditional curriculum that are broadly useful, even while including
topics such as spreadsheets, data analysis, and statistical quality control
that are hardly ever part of high school mathematics. In functional
mathematics, utility is center stage.

The
elements of functional mathematics can be embedded in many different curricula.
However, any mathematics curriculum designed on functional grounds--whether
organized around external contexts or mathematical themes--will emphasize
authentic applications from everyday life and work. In such a curriculum,
students will gain considerable experience with mathematical tasks that are
concrete yet sophisticated, conceptually simple yet cognitively complex. By
highlighting the rich mathematics embedded in everyday tasks, this approach (in
contrast to traditional "vocational math") can dispel both minimalist views
about the mathematics required for work and elitist views of academic
mathematics as an area with little to learn from work-based problems.

A
curriculum based on functional mathematics requires appropriate content,
authentic contexts, engaging tasks, and active learning. By featuring
mathematics in common contexts, a functional curriculum can motivate students
to link meaning with mathematics. Such contexts invite variations that can
propel students to deep understanding and stimulate mathematical habits of mind.

Functional
mathematics channels the much-criticized "mile-wide, inch-deep" curricular
river into a narrower stream of ideas and procedures that reinforce each other
as students progress through school and college and on into careers. It
provides a rich foundation of experience and examples on which students can
build subsequent abstractions and generalizations. Indeed, to fulfill its
goals, a functional curriculum must leave students well prepared not only for
work but also for subsequent courses in more abstract mathematics.

Functional
mathematics addresses many needs that are otherwise often neglected. Neither
traditional college-preparatory mathematics curricula nor the newer
standards-inspired curricula were designed specifically to meet either the
technical and problem-solving needs of the contemporary workforce or the modern
demands of active citizenship. Although each includes much that is of value for
employment and citizenship, neither provides the context, motivation, or
balance of mathematical topics necessary for employees or citizens in a
data-drenched world.

Indeed,
for most students, functional mathematics will provide better preparation than
current high school curricula. Functional mathematics stresses an in-depth
understanding of topics that are most likely to be used by large numbers of
people. By employing concrete methods in settings that are both complex and
realistic, functional mathematics pushes all students to draw on the full
breadth of mathematics. In short, by focusing on what is useful, functional
mathematics increases both student interest and student learning.

Beyond Eighth Grade:
Functional Mathematics
for Life and Work

For
most of this century, schools have been organized to prepare some students for
college and others for work. But the world of work is changing, much of it
requiring increased use of data and computers, measurements and graphs. In the
twenty-first century, most jobs and all careers will require some form of
postsecondary education. Thus, all students, regardless of career goals, will
benefit from a curriculum that prepares them for both work and higher
education. Instead of either/or, schools must now focus on both/and.

In
this paper, we propose a three-year high school program based on functional
mathematicsthat prepares students for life and work. Typically, classroom applications are
designed to serve mathematics--to offer context, to illustrate use, to motivate
new concepts, or to integrate topics. In functional mathematics, the priorities
are reversed. Instead of applications being introduced to help students learn a
predefined body of knowledge, the mathematical topics are selected to prepare
students to cope with common problems they will face in life and work. Examples
of such real-life problems appear throughout this paper to illustrate in a
concrete way the nature of functional mathematics.

Functional
mathematics, if thoughtfully and rigorously developed, can provide a strong
background both for students entering the workforce and for those moving
directly into postsecondary education. Indeed, for most students, functional
mathematics will provide better preparation than the current high school
curricula. Functional mathematics stresses an in-depth understanding of
fundamental topics that are most likely to be used by large numbers of people.
By employing concrete tools in settings that are both complex and realistic,
functional mathematics pushes students to draw on the full breadth of
mathematics. In short, focusing on useful mathematics increases total learning.

Given the complete record of a soccer league that ended in a three-way tie, devise a fair means of determining which team should be crowned champion.

Need and Urgency

Despite
mathematics' reputation as an ancient subject consisting of indisputable facts,
mathematics education has recently become the source of passionate public
debate. At stake is nothing less than the fundamental nature of school
mathematics: its content (what should be taught), pedagogy (how it should be
taught), and assessment (what should be expected). At times, these "math wars"
have become so heated that Education Secretary Richard Riley (1998b) has issued
a public call for a truce.

At
the risk of oversimplifying, this debate can be characterized as a clash
between "traditionalists" who expect schools to provide the kind of
well-focused mathematics curriculum that colleges have historically expected
and "reformers" who espouse a broader curriculum that incorporates uses of
technology, data analysis, and modern applications of mathematics. The reform
approach is championed by the National Council of Teachers of Mathematics,
whose standards (NCTM, 1989) advocate a robust eleven-year core curriculum for
all students, with supplementary topics for those who are "college-intending."
Critics argue, however, that the NCTM standards are diffuse and ambiguous
(Cheney, 1997; Raimi & Braden, 1998), that they are based on questionable
assumptions about how students learn (Anderson, Reder, & Simon, 1997), and
that curricula based on these standards will not provide the kind of rigorous
preparation students need to succeed in calculus and other college-level
mathematics courses (Wu, 1997).

Largely
left out of this debate is a major constituency of mathematics education:
employers. In today's information age, economic prosperity--for individuals as
well as for the nation--depends on "working smarter, not just working harder."
Yet, a majority of America's businesses report deficiencies in the technical
and problem-solving skills of their workers and a severe shortage of
prospective employees with these requisite skills. The cost of finding skilled
employees has become a serious impediment to growth in many sectors of the U.S.
economy (Carnevale, 1998).

Also
left out are the voices of democracy and citizenship which were so important in
the development of public education in the United States. Now, two centuries
later, quantitative literacy is every bit as important as verbal literacy for
informed participation in civic affairs. Today's news is not only grounded in
quantitative issues (e.g., budgets, profits, inflation, global warming, weather
probabilities) but is also presented in mathematical language (e.g., graphs,
percentages, charts).

Neither
traditional college-preparatory mathematics curricula nor the newer
standards-inspired curricula were designed specifically to meet either the
technical and problem-solving needs of the modern workforce or the modern
demands of active citizenship. Although each includes much that is of value for
citizenship and employment, neither provides the context, motivation, or
balance of mathematical topics necessary for citizens or prospective employees
in a data-drenched world.

The
common curricular alternative--vocational or consumer mathematics--is
significantly worse. Historically, vocational mathematics has provided only a
narrow range of skills limited to middle school topics and devoid of conceptual
understanding (National Center for Education Statistics [NCES], 1996). Such
programs leave students totally unprepared--not only for modern work and
postsecondary education, but even for advanced secondary school mathematics.
Although some innovative school-to-career programs are seeking to change this
pattern of low expectations, the vast majority of secondary schools in the
United States offer students no effective option for mathematics education that
meets the expectations of today's high-performance workplace.

A student plans to take out a $10,000 loan at 7% interest with monthly payments of $120, but before she closes the deal, interest rates rise to 7.5%. What will happen if she keeps her monthly payments at $120?

Employment and Education

Mathematics
is the key to many of the most secure and financially rewarding careers in
every sector of the economy (Business Coalition for Education Reform, 1998).
The impact of computers and information technology can be seen not just in
engineering and science, but in such diverse areas as manufacturing and
agriculture, health care and advertising. To be prepared for careers in
virtually any industry, and especially for changing careers during a lifetime,
secondary school students need to learn a substantial core of mathematics.
However, this core is like neither the abstract pre-engineering mathematics of
the academic curriculum nor the restricted topics of the discredited
"vocational math." New approaches are needed to meet today's challenges.

A
recent survey of 4,500 manufacturing firms revealed that nearly two out of
three current employees lack the mathematics skills required for their work,
and that half lack the ability to interpret job-related charts, diagrams, and
flowcharts (National Association of Manufacturers, 1997). Other reports cite a
major shortage of qualified candidates for jobs in the information technology
industries (Information Technology Association of America, 1997), as well as
for technicians and licensed journeymen in the skilled trades (Mathematical
Sciences Education Board, 1995). Even office work has changed, so that
technical skills are now at a premium (Carnevale & Rose, 1998).

What
current and prospective employees lack is not calculus or advanced algebra, but
a plethora of more basic quantitative skills that could be taught in high
school but are not (Murnane & Levy, 1996; Packer, 1997). They need
statistics and three-dimensional geometry, systems thinking and estimation
skills. Even more important, they need the disposition to think through
problems that blend quantitative data with verbal, visual, and mechanical
information; the capacity to interpret and present technical information; and
the ability to deal with situations when something goes wrong (Forman &
Steen, 1998).

Business
has discovered, and research confirms, that diplomas and degrees do not tell
much about students' actual performance capabilities. For example, data from
the National Assessment of Educational Progress (NAEP) (1997b) show that
twelfth-grade students at the 10th percentile are essentially similar to
fourth-grade students at the 80th percentile. Indeed, the level that NAEP
considers "advanced," and which is achieved by only
8%
of U.S. students, is considered just barely adequate in the context of college
expectations (NAEP, 1997a). Enrollment data for postsecondary mathematics
courses confirm this discrepancy (Loftsgaarden, Rung, & Watkins, 1997):
three out of every four students enrolled in college mathematics courses are
studying subjects typically taught in high school or even middle school (see
Figure 1). Clearly, covering mathematics in school is no guarantee of mastering
it for later use.

1995 Postsecondary Mathematics Enrollments

Figure 1

Nearly
two-thirds of high school graduates enter postsecondary education primarily in
order to obtain further skills and an advanced degree. Unfortunately, fewer
than half of those who begin college attain any degree at all within five
years. Furthermore, the majority of those who begin a traditional liberal arts
program never finish. Although the economy clearly needs employees with
advanced technical training (Judy & D'Amico, 1997), these students--the
majority--end up with just a list of courses and no degree or job certification
(Barton, 1997).

Ever
since the publication of A
Nation at Risk(National Commission on Excellence in Education, 1983), many advocates of
educational reform have built their case on international competitiveness: to
compete in a global economy that is increasingly technological, U.S. workers
need better technical education (Commission on the Skills of the American
Workforce, 1990). Yet, data from international comparisons such as the Third
International Mathematics and Science Study (TIMSS) show that U.S. students are
far from competitive (NCES, 1998). Thus, according to this argument, to remain
internationally competitive, we need to radically overhaul mathematics and
science education (Riley, 1998a).

In
fact, the U.S. economy is thriving despite consistently weak performances by
students on both national and international tests. This paradox has led some
observers to suggest that the problem with weakness in school mathematics and
science education is not so much that it hurts the overall economy, but that it
increases economic inequities by providing the means to a good livelihood to
only a few, primarily those from upper socioeconomic backgrounds (Barton, 1997;
Bracey, 1997). From this perspective, the primary rationale for improving
school mathematics is not competitiveness, but equity: in today's data-driven
world, there is no justification for approaches to mathematics education that
filter out those with greatest need and equip only the best-prepared for
productive high-income careers.

A
high school curriculum that helps all students master functional mathematics
would effectively address issues of both equity and competitiveness. Since all
students would study the same curriculum, all would have equal opportunity to
master the mathematics required for the new world of work. Moreover, a
three-year core of functional mathematics would give all students a strong
platform on which to build either technical work experience or advanced
education. Either route would lead to productive careers.

A large load of topsoil forms a conical pile. Because of its size, you cannot directly measure either its diameter or its height. Find a strategy for estimating its volume.

Theory and Practice

Historically,
education in the United States has vacillated between the liberal and the
pragmatic, between Robert Maynard Hutchins and John Dewey. Mathematics reflects
a similar tension in the delicate balance of theory and practice, of the pure
and the applied (Thurston, 1990). Through most of this century, school
mathematics has oscillated back and forth between these poles (Kilpatrick,
1997). Indeed, nearly a century ago, the president of the American Mathematical
Society lamented the "grievous" separation of pure from applied mathematics and
urged schools to provide a more "practical" mathematics education: "With the
momentum of such [education], college students would be ready to proceed
rapidly and deeply in any direction in which their personal interests might
lead them" (Moore, 1903). Today's effort to make mathematics more functional
for all students is just the latest chapter in this long saga.

In
recent years, this debate has been expressed in the form of standards, both
academic and occupational. Coordinating these standards will involve not only
issues of content and pedagogy, but also the balance of school-based vs.
work-based learning (Bailey, 1997). Historically, vocational curricula designed
to prepare students for work have been burdened by second-class status in
comparison with more rigorous academic curricula. Too often, vocational
programs became dumping grounds for students who appeared slow or
unmotivated--"other people's children." Most programs responded by limiting
goals and lowering expectations, thereby offering stunted education to students
who were already behind. In contrast, contemporary career-oriented curricula
have been designed not primarily as training for low-skill jobs but as
motivation for rigorous study, both academic and vocational (Bailey &
Merritt, 1997; Hoachlander, 1997). By setting high standards, these programs
offer significant responses to the twin challenges of equity and competitiveness.

Mathematics
provides a microcosm of the duality between the academic and the vocational.
Widely perceived as the epitome of theory and abstraction, mathematics is also
valued as a powerful, practical tool (Odom, 1998). In many occupations,
quantitative literacy is as important as verbal literacy (Steen, 1997);
however, if mathematics education is to serve the world of work, a different
type of experience than that found in typical mathematics courses is required
(National Research Council, 1998).

Between
theory and application lies professional practice--the synthesis of thought and
action employed by practitioners in all vocations. Many have argued that
practice, properly understood, can be a legitimate and unifying goal of
education. Practice is functional knowledge, the kind of know-how that allows
people to get things done. According to educator Lee Shulman (1997), practice
can provide a context in which theory becomes meaningful, memorable, and
internalizable. Peter Denning (1997), a computer scientist, believes that
practice--not knowledge or literacy--is what constitutes true expertise.
Indeed, practice is what people tend to expect of schools, especially of
mathematics education. It is at the heart of functional mathematics.

An
infusion of practice into school mathematics can overcome what Shulman (1997)
identifies as major deficiencies of theoretical learning: loss of learning ("I
forgot it"), illusion of learning ("I thought I understood it"), and
uselessness of learning ("I understand it but I can't use it"). Adults who are
not professional users of mathematics will recognize these deficiencies from
their own experiences. Little of what adults learned in school mathematics is
remembered or used, so the accomplishment of "learning" mathematics is often an
illusion. In fact, the mathematics many students are force-fed in traditional
school environments creates a severe psychological impediment to the practice
of mathematics in adult life (Buxton, 1991; Cockroft, 1982). Functional
mathematics avoids many of these pitfalls by emphasizing that the goal of
mathematics education is not just mathematical theory and word problems, but
authentic mathematical practice.

Habitat for Humanity uses volunteer labor to build inexpensive homes, which it sells for the cost of materials. Using information on standard building supplies obtained from a local lumberyard, design a simple home whose building materials can be obtained for $15,000.

High School Mathematics

Traditionally,
high school mathematics has served two different purposes--to prepare
college-intending students for calculus (and other mathematics-based courses)
and to equip other students with necessary skills, mostly arithmetic, so that
they can function as employees, homemakers, and citizens. Although most
traditionalists--and most parents and grandparents--still support these dual
goals, reformers argue for a common curriculum for allstudents which emphasizes problem solving, communication, reasoning, and
connections with other disciplines.

Proposed
goals for school mathematics can be found in many sources. Some focus directly
on K-12, others on the needs of postsecondary education or employers. NCTM
(1989) provides a comprehensive set of standards for grade levels K-4, 5-8, and
9-12 that represents the "reform" perspective. In contrast, California recently
adopted mathematics standards that represent a more traditional perspective
(California Academic Standards Commission, 1997). The American Mathematical
Association of Two-Year Colleges (1995) articulated standards for college
mathematics before calculus that include expectations for the mathematical
foundation that students need to succeed in college. In addition, in the
influential report What
Work Requires of Schools(Secretary's Commission on Achieving Necessary Skills [SCANS], 1991), the U.S.
Department of Labor outlined both foundation skills and broad employability
competencies for mathematics and other subjects.

These
standards differ greatly in both mathematical content and rhetorical style (see
Appendix A), although most have overlapping goals. Indeed, to succeed in the
real world of teachers and parents, schools and school boards, a mathematics
curriculum must

(1)
meet society's expectations of what all high school graduates should know and
be able to do.

(2)
reflect priorities common to state and national guidelines.

(3)
increase the number of students who successfully persist in advanced
mathematics-based courses, including calculus.

(4)
enable students to see and use mathematics in everyday aspects of life and work.

(5) help
students understand and use correct mathematical language.

Functional
mathematics must also meet these objectives. The first two objectives establish
priorities: to focus early and often on what everyone agrees must be learned,
leaving to later (or to optional strands) those topics that only some students
will find interesting or important. The third objective establishes a standard
of quality: to increase the number of students who persist in further
mathematics-based courses (including calculus, the traditional hallmark of
mathematical success). The fourth objective conveys a commitment to utility--to
ensure that students see mathematics as something real in their lives rather
than as an alien subject encountered only in school. Finally, the fifth
objective stresses command of the language of mathematics, a skill at least as
important for success as a command of English.

By
meeting these objectives, functional mathematics will satisfy the general
public's expectations of school mathematics. In addition, these objectives also
enhance functional mathematics' primary goal of preparing students for life and
work. Consistent quality and high standards are essential in today's
high-performance industries. Persistence in mathematics is not just of academic
importance; it is also one of the best predictors of success in careers
(Commission on the Skills of the American Workforce, 1990). Moreover, the
language of mathematics provides the power to analyze and express complex
issues in all aspects of life and work. Fluency in this language is important
not only for productive employees but also for careful consumers and critical
citizens.

In
functional mathematics, utility is center stage. Other objectives play
important but supporting roles. Unfortunately, many mathematicians and
mathematics teachers find utility at best a bleak justification (Howe, 1998)
for a subject that they chose for its beauty and elegance. For them, the power
of mathematics--in Eugene Wigner's famous phrase, its "unreasonable
effectiveness"--is not its primary virtue, but merely a consequence of its
elegance and internal structure. Thus, mathematicians are wont to stage their
subject with theory and abstraction at the center, employing applications,
technology, and practice as needed to help promote understanding.

To
engage mathematicians and mathematics teachers, functional mathematics needs to
be seen in terms of both utility and beauty. For many students, utility can be
a path to beauty, while for others, mathematics by itself provides sufficient
internal motivation to sustain interest and accomplishment. For any mathematics
curriculum to succeed with all students, it must build on the twin foundations
of utility and elegance.

What measurements do you need to take in order to tile the floor of a room? How can you use these measurements to determine the number of regular tiles, border tiles, and corner tiles that are needed? What if you decide to lay the main tiles on a 45° angle?

Functional Mathematics

Functional
mathematics comprises content, curriculum, context, and pedagogy. By content we
simply mean the mathematics students should know and be able to do after
finishing the first three years of high school mathematics (see Appendix B).
Because mathematics is mathematics--whether traditional, reform, or
functional--most of these elements are unsurprising. Although some topics are
uncommon (e.g., index numbers, tolerances, three-dimensional geometry, normal
curve, quality control charts, standards of proof, financial mathematics,
spreadsheets), most are taught in any high-quality high school mathematics
program. To achieve its goal of preparing students for both work and further
education, functional mathematics respects the many parts of the traditional
curriculum that are broadly useful, even while reshaping the boundaries to
reflect its distinctive objectives.

The
outline of functional mathematics in Appendix B reflects an inventory of
mathematical topics selected for their importance in daily life and modern jobs
as well as for their value in providing a strong foundation for further
education. This outline is organized in predictable strands that cover what is
normally subsumed under the umbrella of mathematics: numbers and data,
measurement and space, growth and variation, chance and probability, reasoning
and inference, variables and equations, modeling and decisions. Real problems
cut across all this mathematics, just as these topics cut across the diverse
contexts of authentic mathematical practice.

Clearly,
many of the elements of functional mathematics are identical to the mathematics
found in both traditional and reformed curricula. The core of school
mathematics is more or less the same, even if viewed (or taught) from different
perspectives. Percentages and ratios; linear and quadratic equations; areas,
angles, and volumes; and exponential growth and trigonometric relations must be
included in any strong high school mathematics program. The distinctions among
traditional, reformed, and functional curricula lie not so much in core content
as in contexts, emphases, and pedagogy.

Nonetheless,
prospective employees for the new high-performance workplace need expertise in
several aspects of mathematics not now emphasized sufficiently in school. On
the one hand, students need greater experience recognizing and using some parts
of middle school mathematics such as ratio, percentage, and measurement
geometry that, although covered in current programs, are not used sufficiently
to be learned well. On the other hand, as prospective employees, they need to
understand and be able to use mathematical notions such as data analysis,
statistical quality control, and indirect measurement that are hardly ever
required in high school (Forman & Steen, 1998).

In
addition to shifting the balance of topics, functional mathematics provides
much greater emphasis on "systems thinking"--on habits of mind that recognize
complexities inherent in situations subject to multiple inputs and diverse
constraints. Examples of complex systems abound--from managing a small business
to scheduling public transportation, from planning a wedding to reforming
social security. At all levels from local to national, citizens, policymakers,
employees, and managers need to be able to formulate problems in terms of
relevant factors and design strategies to determine the influence of those
factors on system performance. Although such systems are often so complex that
they obscure the underlying mathematics, the skills required to address
realistic problems very often include many that are highly mathematical.

A
curriculum built on functional mathematics gives students many opportunities to
solve realistic problems and build mathematical understanding. Nevertheless, to
make this learning valuable for work and further education--as well as to
enhance understanding--such a curriculum must also help students become fluent
in the language of mathematics. Individuals need to be able to read,
understand, and interpret technical material with embedded charts and diagrams
(e.g., property tax bills, stock market reports); they need to be able to speak
clearly about mathematical ideas (e.g., as a salesman explaining the interest
and payoff on an insurance policy); and they need extensive experience writing
reports based on mathematical and technical analysis (e.g., a recommendation to
a supervisor summarizing the conclusion of a study).

Functional
mathematics channels the much-criticized "mile-wide, inch-deep" curricular
river into a narrower but deeper stream of ideas and procedures that reinforce
each other as students progress through school and college and on into careers.
It provides a rich foundation of experience and examples on which students can
build subsequent abstractions and generalizations. Indeed, to fulfill its
goals, a functional curriculum must leave students well-prepared not only for
work but also for subsequent courses in more abstract mathematics.

You are helping your brother-in-law build a garage on gently sloped land next to his house. After leveling the land, you begin staking out the foundation. To check that corners are square, you measure the diagonals and discover that they differ by 3 inches. Is that because the corners may not be perfectly level, or because they are not perfectly square? How can you determine what needs fixing to make sure that you start with a foundation that is both level and square?

Functional Curricula

The
elements of functional mathematics can be embedded in many different
curricula--the paths students follow through their education. Although some
parts of mathematics impose a necessary order on the curriculum (e.g.,
arithmetic before algebra; linear equations before quadratic), large parts of
mathematics can be approached from many different directions. Data analysis can
be either a motivation for or an application of graphing and algebra, geometry
can either precede or follow algebra; and each can reinforce the other. The
order in which elements are listed bears no relationship to the order in which
they may be taught through a three-year core curriculum.

The
elements of functional mathematics arise from common contexts of life and
work--measuring objects, managing money, scheduling time, making choices, and
projecting trends. Although it is possible to organize a curriculum around such
contexts, without a list of elements such as those in Appendix B to guide
instruction, the mathematics itself may remain largely hidden. Alternatively, a
functional curriculum can be organized around mathematical themes such as the
sections of Appendix B. Indeed, the latter fits better the experience of most
mathematics teachers and is more likely to be adaptable to most school settings.

Any
mathematics curriculum designed on functional grounds--whether organized around
external contexts or mathematical themes--will emphasize authentic applications
from everyday life and work. In such a curriculum, students will gain
considerable experience with mathematical tasks that are concrete yet
sophisticated, conceptually simple yet cognitively complex (Forman & Steen,
1995). A functional curriculum compels a better balance of statistics
(numbers), geometry (space), and algebra (symbols)--the three major branches of
the mathematical sciences. By highlighting the rich mathematics embedded in
everyday tasks, this approach (in contrast to traditional "vocational math")
can dispel both minimalist views about the mathematics required for work and
elitist views of academic mathematics as an area with little to learn from
work-based problems (Bailey & Merritt, 1997; Forman & Steen, 1998).

Because
of the history of low standards in traditional vocational programs, many
teachers and parents believe that a work-focused curriculum will necessarily
lack the rigor of a precollege academic track. Contrary to this belief, the
"zero-defect" demands of the high-performance workplace for exacting standards
and precise tolerances actually impose a much higher standard of rigor than do
academic programs that award students a B for work that is only 80-85%
accurate. Moreover, the lengthy and subtle reasoning required to resolve many
problems that arise in real contexts provides students with experience in
critical thinking that is often lacking in academic courses that rush from
topic to topic in order to cover a set curriculum.

Five friends meet for dinner in a restaurant. Some have drinks and others do not; some have dessert and others do not; some order inexpensive entrees, others choose fancier options. When the bill comes they need to decide whether to just add a tip and split it five ways, or whether some perhaps should pay more than others. What is the quickest way to decide how much each should pay?

Teaching Functional Mathematics

Although
the public thinks of standards primarily in terms of performance expectations
for students, both the mathematics standards
(NCTM,
1989) and the science standards (National Research Council, 1996) place equal
emphasis on expectations for teaching, specifically that it be active,
student-centered, and contextual:

Active
instructionencourages students to explore a variety of strategies; to make hands-on use of
concrete materials; to identify missing information needed to solve problems;
and to investigate available data.

Student-centered
instructionfocuses on problems that students see as relevant and interesting; that help students learn to work with others; and that strengthen students' technical communication skills.

Contextual
instructionasks students to engage problems first in context, then with mathematical formality; suggests resources that might provide additional information; requires that students verify the reasonableness of solutions in the context of the original problem; and encourages students to see connections of mathematics to work and life.

These
expectations for effective teaching are implicitly reinforced in recently
published occupational skill standards (National Skill Standards Board, 1998)
that outline what entry-level employees are expected to know and be able to do
in a variety of trades. Although these standards frequently display performance
expectations for basic mathematics as lists of topics, the examples they
provide of what workers need to be able to do are always situated in specific
contexts and most often require action outcomes (Forman & Steen, in press).

Most
students learn mathematics by solving problems. In traditional mathematics
courses, exercises came in two flavors: explicit mathematical tasks (e.g.,
solve, find, calculate) and dreaded "word problems" in which the mathematics is
hidden as if in a secret code. Indeed, many students, abetted by their
teachers, learn to unlock the secret code by searching for key words (e.g., lessfor
minus, totalfor plus)
rather than by thinking about the meaning of the problem (which may be a good
thing, because so many traditional word problems defy common sense).

In
a curriculum focused on functional mathematics, tasks are more likely to
resemble those found in everyday life or in the workplace than those found in
school textbooks. Students need to think about each problem afresh, without the
clues provided by a specific textbook chapter. Rather than just being asked to
solve an equation or calculate an answer, students are asked to design, plan,
evaluate, recommend, review, define, critique, and explain--all things they
will need to do in their future jobs (as well as in college courses). In the
process, they will formulate conjectures, model processes, transform data, draw
conclusions, check results, and evaluate findings. The challenges students face
in a functional curriculum are often nonroutine and open-ended, with solutions
taking from minutes to days, and requiring diverse forms of presentation (oral,
written, video, or computer). As in real job situations, some work is done
alone, and some in teams.

A round chimney 8" in diameter protrudes from a roof that has a pitch of 3:1. Draw a pattern for an aluminum skirt that can be cut out of sheet metal and bent into a cone to seal the chimney against rain.

Mathematics In Context

Students'
achievements in school mathematics depend not only on the content of the
curriculum and the instructional strategies employed by the teacher but also on
the context in which the mathematics is embedded. Traditionally, mathematics
has served as its own context: as climbers scale mountains because they are
there, so students are expected to solve equations simply because it is in the
nature of equations to be solved. From this perspective, mathematics is
considered separate from and prior to its applications. Once the mathematics is
learned, it supposedly can then be applied to various problems, either
artificial or real.

Many
of the new curricula developed in response to the NCTM standards or state
frameworks give increased priority to applications and mathematical models. In
some of these programs, applications are at the center, providing a context for
the mathematical tools prescribed by the standards. In others, applications
serve more to motivate topics specified in the standards. In virtually all
cases, the applications found in current curricula are selected, invented, or
simplified to serve the purpose of teaching particular mathematical skills or
concepts. In contrast, the mathematical topics in a functional curriculum are
determined by the importance of the contexts in which they arise.

Devise criteria and procedures for fair addition of a congressional district to a state in a way that will minimize disruption of current districts while creating new districts that are relatively compact (non-gerrymandered) and of nearly equal size.

Employing Computers

It
has been clear for many years that technology has changed priorities for
mathematics. Much of traditional mathematics (from long division to integration
by parts) was created not to enhance understanding but to provide a means of
calculating results. This mathematics is now embedded in silicon, so training
people to implement these methods with facility and accuracy is no longer as
important as it once was. At the same time, technology has increased
significantly the importance of certain parts of mathematics (e.g., statistics,
number theory, discrete mathematics) that are widely used in information-based
industries.

Many
mathematics teachers have embraced technology, not so much because it has
changed mathematics but because it is a powerful pedagogical tool. Mathematics
is the science of patterns (Devlin, 1994; Steen, 1988), and patterns are most
easily explored using computers and calculators. Technology enables students to
study patterns as they never could before, and in so doing, it offers
mathematics what laboratories offer science: a source of evidence, ideas, and
conjectures.

The
capabilities of computers and graphing calculators to create visual displays of
data have also fundamentally changed what it means to understand mathematics.
In earlier times, mathematicians struggled to create formal symbolic systems to
represent with rigor and precision informal visual images and hand-drawn
sketches. However, today's computer graphics are so sophisticated that a great
deal of mathematics can be carried out entirely in a graphical mode. In many
ways, the medium of computers has become the message of mathematical practice.

Finally,
and perhaps most significantly, computers and calculators increase dramatically
the number of users of mathematics--many of whom are not well-educated in
mathematics. Previously, only those who learned mathematics used it. Today,
many people use mathematical tools for routine work with spreadsheets,
calculators, and financial systems--tools that are built on mathematics they
have never studied. For example, technicians who diagnose and repair electronic
equipment employ a full range of elements of functional mathematics--from
number systems to logical inferences, from statistical tests to graphical
interpretations. Broad competence in the practice of technology-related
mathematics can boost graduates up many different career ladders.

This
poses a unique challenge for mathematics education: to provide large numbers of
citizens with the ability to use mathematics-based tools intelligently without
requiring that they prepare for mathematics-based careers. Although
mathematicians take for granted that learning without understanding is
ephemeral, many others argue that where technology is concerned, it is more
important for students to learn how to use hardware and software effectively
than to understand all the underlying mathematics. But even those who only use
the products of mathematics recognize the value of understanding the underlying
principles at a time when things go wrong or unexpected results appear. In a
functional curriculum where, for example, algebra emerges from work with
spreadsheets, the traditional distinction between understanding and competence
becomes less sharp.

-- How many school teachers are there in New York City? How many electricians? How many morticians?
-- How many words are there in all the books in the school library? How many megabytes of disk storage would be required to store the entire library on a computer?

Avoiding Pitfalls

Those
who develop materials and examples for a functional curriculum need to avoid
some common pitfalls that plague all attempts at situating mathematics in
authentic contexts. On the one hand, there is the temptation to give priority
to the mathematics, either by selecting tasks to ensure coverage of
mathematical topics without much regard to the tasks' intrinsic importance or
by imposing unwarranted structure on a contextually rich problem in the
interest of ensuring appropriate mathematical coverage. On the other hand, it
is easy to overlook interesting mathematics hidden beneath the surface of many
ordinary tasks or to choose problems that fail to help students prepare for
advanced study in mathematics. Any curriculum that is to prepare students for
subsequent mathematics-dependent courses must recognize the importance of
intellectual growth and conceptual continuity in the sequencing of tasks in
which mathematical activities are embedded.

Context-rich
mathematics curricula often present tasks in the form of worksheets, outlining
a series of short-answer steps that lead to a solution. While ostensibly
intended to help students organize their thinking and assist teachers in
following students' work, these intellectual scaffolds strip tasks of
everything that makes them problematic. Indeed, worksheets reveal a didactic
posture of traditional teaching (teacher tells, students mimic) that undermines
learning and limits understanding. Students will learn and retain much more
from the chaotic process of exploring, defending, and arguing their own
approaches.

Finally,
although a functional mathematics curriculum is motivated largely by examples
that seem to lie outside the world of mathematics, it is nonetheless very
important for students' future study that instructors bring mathematical
closure at appropriate points. Students need to recognize and reflect on what
they have learned; to be clear about definitions, concepts, vocabulary,
methods, and potential generalizations; and to have sufficient opportunity to
reflect on the accomplishments and limitations of mathematics as a tool in
helping solve authentic problems.

A patient with an aggressive cancer faces two options for treatment: With Option A, he has a 40% chance of surviving for a year, but if he makes it that long then his chance of surviving a second year is two out of three. With Option B, he has a 50-50 chance of surviving each of the first and second years. Survival rates beyond the second year are similar for each option. Which choice should he make?

Mathematics in Life and Work

The
diverse contexts of daily life and work provide many realistic views of
functional mathematics--of the mathematical practice underlying routine events
of daily life. These contexts offer episodic views, incomplete in scope and
less systematic than a list of elements, but more suggestive of the way
functional mathematics may be introduced in courses.

Reading Maps.
Road maps of cities and states provide crucial information about routes and
locations. For those who know how to "read" them, maps also convey scale and
direction, helping drivers know which way to turn at intersections, permitting
quick estimates of driving time, and revealing compass directions that relate
to highway signs at road intersections. Map scales are just ratios--an
essential part of school mathematics. Different scales not only convey
different detail, but also require different translations to represent distance.

Reading
maps is not just a matter of thinking of distances in different scales. In many
cases, the geometry of maps represents other features such as temperature or
soil content. Most common are weather maps with color-coded regions showing
gradations in recorded or predicted temperatures. Similar maps sometimes
display recorded or predicted precipitation, barometric pressure, vegetation
features, or soil chemistry. Like topographic maps used by hikers, these maps
represent some feature of the landscape that changes from place to place. The
spacing between regions of similar temperature (or pressure, or elevation)
conveys the steepness (or gradient) of change--what mathematicians call the
"slope" of a line.

Scale-drawings
and blueprints are also widely used to illustrate details of homes, apartments,
and office buildings. These drawings represent sizes of rooms, locations of
windows and doors, and--if the scale permits--locations of electrical outlets
and plumbing fixtures. Architects' rulers with different units representing one
foot of real space make it possible to read real distances off scale drawings,
taking advantage of the geometrical properties of similar figures. New
geographic information systems (GIS) encode spatially oriented data in a form
suitable for computer spreadsheets, thereby enabling other factors (e.g.,
costs, environmental factors) to be logically linked to the geometric structure
of a map.

Ensuring Quality.
Statistical process control (SPC) and statistical quality control (SQC) are
crucial components of high-performance manufacturing, where "zero defect" is
the goal. Instead of checking and repairing products after manufacture, firms
like Boeing, General Motors, Kodak, Motorola, and Siemens now insist that at
every step in the manufacturing process, materials, parts, and final products
be manufactured within tight tolerances. Moreover, workers on assembly lines
are responsible for ensuring this consistent level of quality.

The
two tools that make this possible are based on statistics--the science of
collecting and organizing data. The first, statistical process control, occurs
during manufacture: assembly line workers chart key indicators of the
process--perhaps the temperature of a mixture or the pitch of a grinding
tool--on graph paper marked with curves representing the limits determined by
the required (or contracted) tolerances. If the process strays outside these
limits, or approaches them too often, workers may decide to shut down the
assembly line to make adjustments in the manufacturing process rather than risk
producing products that do not meet design specifications.

Statistical
quality control is like statistical process control, but takes place when
components (e.g., computer chips) are completed. By sampling finished products
and charting their performance characteristics, workers can identify potential
problems before products exceed permitted tolerances--and then take action to
prevent the shipping or further manufacture of defective (i.e.,
out-of-tolerance) products.

Using Spreadsheets.
Almost everyone who works with a computer uses a word processor for writing,
whether for correspondence or business reports. Almost as popular are "number
processors," commonly known as spreadsheets. Originally designed as a tool for
accountants, spreadsheets are ubiquitous both in the office and at
home--wherever anyone deals with budgets and expenses, taxes and investments.
Spreadsheets are used to record business inventories and scientific data, to
keep track of medical records and student grades, to organize crop records and
airline schedules. Virtually any systematic information can be made more useful
by being put in a properly organized spreadsheet.

To
a mathematician, a spreadsheet is just algebra playing on a popular stage. The
basic operations of a spreadsheet--adding cells together, calculating
percentages, projecting growth rates, determining present values--are entered
as formulas into the appropriate cells. More complex formulas (e.g.,
exponential, financial, trigonometric) are available from a pull-down menu.
Once the computations are completed, the results can be displayed in graphs of
various sorts (lines, bars, pies), often in vivid color.

Figuring
out how to translate a task into a spreadsheet design is just like setting up a
word problem in algebra: it involves identifying important variables and the
relations among them. Preparing a spreadsheet requires equations which are
suitably located in the cells. The spreadsheet does the arithmetic, and the
designer does the algebra. Then, as in any mathematical exercise, the designer
needs to check the results--typically by specifying independent computations to
confirm key spots in the spreadsheet. (For example, adding all the entries in a
grid can confirm the accuracy of the sum of the row totals, thus catching
possible errors in the spreadsheet formulas.) Variables, equations, graphs,
word problems--the ingredients of a good algebra course--are just the ticket
for mastering spreadsheets.

Building Things.
One in every four American workers builds things--automobiles or airplanes,
bicycles or buildings, containers or chips. These products are
three-dimensional, created by casting and cutting, by folding and fastening, by
molding and machining. Designing things to be built (the work of engineers and
architects) and building objects as designed (the work of carpenters and
machinists) require impressive feats of indirect measurement, three-dimensional
geometry, and visual imagination.

In
a typical aluminum airplane part, for example, some measurements are specified
by the designers, while others must be calculated in order to program the
cutting tool that will actually create the part. In three dimensions, things
are even more complicated. Planning how to drill holes at specified angles in a
block of aluminum whose base is not square and whose sides are tilted in odd
directions would tax the skills of most mathematics teachers. But machinists
are expected to perform these calculations routinely to determine settings on a
"sine plate," a device whose surface can tilt in two different dimensions in
order to compensate for odd angles on the part that is to be drilled.

Both
designers and builders now use computer-assisted design (CAD) and
computer-assisted manufacturing (CAM) to ensure the exacting tolerances
required for high-performance manufacturing. To use these tools effectively,
workers need to have mastered the basic skills of drawing geometric objects,
measuring distances, and calculating angles, distances, areas, and volumes. The
basic principles of geometry in three dimensions are the same as those in two
dimensions, but the experience of working in three dimensions is startlingly
more sophisticated. A good command of geometry and trigonometry is essential
for anyone building things in today's manufacturing industries.

Thinking Systemically.
Systems surround us--in commerce, science, technology, and society. In complex
systems, many factors influence performance, thus making the task of solving
problems inherently multidimensional. Indeed, the interaction of different
factors is often difficult to predict, sometimes even counterintuitive. Complex
systems defy simplistic single answers. Thus, the first step in mathematical
analysis is often to prepare an inventory of all possible factors that might
need to be considered.

For
example, the rise of efficient package delivery services and instantaneous
computer communication have enabled many manufacturing companies to operate
with minimum inventories, thus saving warehousing costs but risking a shutdown
if any part of the network of suppliers fails. Understanding how a system of
suppliers, communication, and transportation works requires analysis of
capacity, redundancy, single-point failures, and time of delivery--all
involving quantitative or logical analyses.

Other
system problems arise within the everyday work of a typical small business. For
example, the stockroom of a shoe store holds several thousand boxes labeled by
manufacturer, style, color, and size and arranged on floor-to-ceiling shelves.
Deciding how to arrange these boxes can have a significant impact on the profit
margin of the store. Obvious options are by manufacturer, by style, by size, by
frequency of demand, or by date of arrival. Clerks need to be able to find and
reshelve shoes quickly as they serve customers. But they also need to be able
to make room easily for new styles when they arrive, to compare regularly the
stockroom inventory with sales and receipt of new shoes, and to locate
misshelved shoes. Mathematical thinking helps greatly in exploring the
advantages and disadvantages of the many possible systems for arranging the
stockroom.

Making Choices.
Life is full of choices--to rent an apartment or purchase a home; to lease or
buy a car; to pay off credit card debt or use the money instead to increase the
down payment on a house. All such choices involve mathematical calculations to
compare costs and evaluate risks. For example:

The
rent on your present apartment is $1,200 per month and is likely to increase 5%
each year. You have enough saved to put a 25% down payment on a $180,000
townhouse with 50% more space, but those funds are invested in an aggressive
mutual fund that has averaged 22% return for the last several years, most of
which has been in long-term capital gains (which now have a lower tax rate).
Current rates for a 30-year mortgage with 20% down are about 6.75%, with 2
points charged up front; with a 10% down payment the rate increases to 7.00%.
The interest on a mortgage is tax deductible on both state and federal returns;
in your income bracket, that will provide a 36% tax savings. You expect to stay
at your current job for at least 5-7 years, but then may want to leave the
area. What should you do?

This
sounds like a problem for a financial planner, and many people make a good
living advising people about just such decisions. But anyone who has learned
high school mathematics and who knows how to program a spreadsheet can easily
work out the financial implications of this situation. Moreover, by doing it on
a spreadsheet, it is quite easy to examine "what if" scenarios: What if the
interest rate goes up to 7% or 7.25%? What if the stock market goes down to its
traditional 10-12% rate of return? What if a job change forces a move after
three years?

In
contrast to many problems of school mathematics which are routine for anyone
who knows the right definitions (e.g., what is cos ([pi]/2)?) but mystifying
otherwise, this common financial dilemma is mathematically simple (it involves
only arithmetic and percentages) but logically and conceptually complex. There
are many variables, some of which need to be estimated; there are many
relationships that interact with each other (e.g., interest rates and tax
deductions); and the financial picture changes each year (actually, each month)
as payments are made.

The
complex sequence of reasoning involved in this analysis is typical of
mathematics, which depends on carefully crafted chains of inferences to justify
conclusions based on given premises. Students who can confidently reason their
way through a lengthy proof or calculation should have no problem being their
own financial advisors. And students who learn to deal with long chains of
reasoning inherent in realistic dilemmas will be well prepared to use that same
logic and careful reasoning if they pursue the study of mathematics in college.

In preparing fertilizer for a garden, a homeowner poured one quart of concentrated liquid fertilizer into a two-gallon can and filled the can with water. Then she discovered that the proper ratio of fertilizer to water should be 1:3. How much more liquid fertilizer should she add to the current mixture to obtain the desired concentration?

Making Mathematics Meaningful

Those
who discuss mathematics education frequently describe mathematical knowledge in
broad categories such as skills and understanding, concepts and facts,
procedures and practices, or insights and knowledge. Whole volumes of
educational research are devoted to distinguishing among these different
aspects of mathematical knowledge. The standards movement has tended to subsume
all these distinctions into two categories of knowledge and performance: what
students should know and what students shoud be able to do (Ravitch, 1995;
Tucker & Codding, 1998).

The
two broad cultures of mathematics education argue with each other largely
because they differ in the interpretations they give to these different aspects
of mathematical knowledge. Those who favor the traditional curriculum centered
on algebra, functions, and Euclidean geometry argue that mastery of facts and
basic skills are a prerequisite to understanding and performance. Reformers who
favor a broader curriculum take a more constructivist view--that understanding
and mastery are an outgrowth of active engagement with contextualized
mathematics. Regardless of approach or emphasis, both traditional and reform
curricula generally cover a similar set of topics designed to move students
along the path from arithmetic to calculus.

Functional
mathematics follows much the same path, with variations that reflect its
grounding in authentic problems. However, by embedding mathematics in practice,
functional mathematics can offer students both theory and know-how. Although in
some technical areas, practical "of-the-moment" learning offers little that
outlasts the next generation of gadgets, the logical structure that unites
mathematics guarantees that all understanding, no matter how specific, has the
potential to enhance mastery of other areas. What matters for long-term mastery
of mathematics is not so much which particular skills are learned as that the
process of learning be, in Shulman's words, "meaningful, memorable, and
internalizable." Although topics in functional mathematics may be chosen for
proximate utility, their study can provide insight and understandings
sufficient for lifelong learning.

A
curriculum based on functional mathematics requires appropriate content,
authentic contexts, engaging tasks, and active instruction. By featuring
mathematics in common contexts, a functional curriculum can motivate students
to link meaning with mathematics. The best problem settings offer opportunities
for exploration from multiple perspectives, including graphical, numerical,
symbolic, verbal, and computational. Technology--from graphing calculators and
word processors to spreadsheets and symbolic algebra systems--can enhance
understanding from each of these perspectives. Effective contexts provide
opportunities for horizontal linkages among diverse areas of life and work as
well as vertical integration from elementary ideas to advanced topics.
Experience with rich contexts helps students recognize that asking questions is
often as important as finding answers. Such contexts invite variations that can
stimulate mathematical habits of mind and propel students to deep understanding.

Devlin,
K. (1994). Mathematics: The science of patterns. New York: Scientific American Library.

Forman,
S. L., & Steen, L. A. (1995). Mathematics for work and life. In I. Carl (Ed.), Seventy-five years of progress: Prospects for school mathematics(pp. 219-241). Reston, VA: National Council of Teachers of Mathematics.

National
Commission on Excellence in Education (NCEE). (1983). A nation at risk: The imperative for educational reform. Washington, DC: U.S. Government Printing Office. Available: http://www.ed.gov/pubs/NatAtRisk/ [May 28, 1999].

Riley,
R. W. (1998b). The state of mathematics education: Building a strong foundation
for the 21st century. Notices of the American Mathematical Society,
45, 487-490. Available: http://www.ams.org/notices/199804/riley.pdf [May 28, 1999].

Wu,
H. (1997). The mathematics education reform: Why you should be concerned and
what you can do. American Mathematical Monthly, 104, 946-954.

Appendix A:
Excerpts From Different Standards

The
diverse ways that different organizations express expectations for mathematics
illustrate a variety of approaches to setting standards. The excerpts that
follow illustrate this variety in the particular case of algebra, the core of
high school mathematics.

From the National Council of Teachers of Mathematics (1989): In
grades 9-12, the mathematics curriculum should include the continued study of
algebraic concepts and methods so that all students can

Derive
and use the quadratic formula to solve any quadratic equation with real
coefficients; graph equations of the conic sections (parabola, ellipse, circle,
hyperbola), identifying key features such as intercepts and axes.

Describe,
extend, and find the nth term of arithmetic, geometric, and other regular series.

From the American Mathematical Association of Two-Year Colleges (1995): The
study of algebra must focus on modeling real phenomena via mathematical
relationships. Students should explore the relationship between abstract
variables and concrete applications and develop an intuitive sense of
mathematical functions. Within this context, students should develop an
understanding of the abstract versions of basic number properties and learn how
to apply these properties. Students should develop reasonable facility in
simplifying the most common and useful types of algebraic expressions,
recognizing equivalent expressions and equations, and understanding and
applying principles for solving simple equations.
Rote
algebraic manipulations and step-by-step algorithms, which have received
central attention in traditional algebra courses, are not the main focus.
Topics such as specialized factoring techniques and complicated operations with
rational and radical expressions should be eliminated. The inclusion of such
topics has been justified on the basis that they would be needed later in
calculus. This argument lacks validity in view of the reforms taking place in
calculus and the mathematics being used in the workplace.

From the Secretary's Commission on Achieving Necessary Skills (1991):Mathematics.
Approaches practical problems by choosing appropriately from a variety of
mathematical techniques; uses quantitative data to construct logical
explanations for real world situations; expresses mathematical ideas and
concepts orally and in writing; and understands the role of chance in the
occurrence and prediction of events.
Reasoning.
Discovers a rule or principle underlying the relationship between two or more
objects and applies it in solving a problem. For example, uses logic to draw
conclusions from available information, extracts rules or principles from a set
of objects or written text, applies rules and principles to a new situation, or
determines which conclusions are correct when given a set of facts and a set of
conclusions.

Appendix B:
Elements Of Functional Mathematics

These
elements outline aspects of mathematics that are important for all students in
their life and work. They emphasize concrete, realistic topics that arise in
common situations in news, sports, finance, work, and leisure. These elements
can be taught through many different curricula ranging from traditional to
reform, from academic to vocational. Students completing any curriculum that
includes these elements would be well-prepared to enter a wide variety of
technical and academic programs, including a one-year precalculus course.

Numbers and Data

Mental
Estimation.Anticipate total costs, distances, times; estimate unknown quantities (e.g.,
number of high school students in a state or city) using proportional
reasoning; order of magnitude estimates; mental checking of calculator and
computer results.

Growth and Variation

Linear
Change.Situations in which the rate of change is constant (e.g., uniform motion);
contrast with examples where change is nonlinear (e.g., distance vs. time for
falling body). Slope as rate of change; slope-intercept equation, with
graphical significance of parameters. Difference between rate of change and
value of the dependent variable.

Exponential
Growth.Situations such as population growth, radioactivity, and compound interest,
where the rate of change is proportional to size; doubling time and half-life
as characteristics of exponential phenomena; symbolic representation (2
n,
10
n);
ordinary and log-scaled graphs.

Normal
Curve.Situations such as distribution of heights, of repeated measurements, and of
manufactured goods in which phenomena distribute in a bell-shaped curve.
Examples of situations in which they do not (e.g., income, grades,
typographical errors, life spans). Parameters and percentages of normal
distribution; z-scores, meaning of 1-, 2-, and 3-s. Area as measure of
probability.

About the Authors

Susan
L. Forman is Professor of Mathematics at Bronx Community College, City
University of New York (CUNY). While on extended leave from the College, she
served as Senior Program Officer for Education at the Charles A. Dana
Foundation (1995-1997) and as Director of College and University Programs for
the Mathematical Sciences Education Board of the National Academy of Sciences
(1992-1995). Previously she served as Coordinator of Academic Computing at the
City University of New York and Program Officer at the Fund for the Improvement
of Postsecondary Education (FIPSE).

Forman
has served as First Vice-President of the Mathematical Association of America
(1992-1994); as Chair of the Association's New York Metropolitan Section
(1997-1999); and as President of the New York State Mathematics Association of
Two-Year Colleges (1985-1986). She received her doctorate in mathematics
education and research from Columbia University.

Lynn
Arthur Steen is Professor of Mathematics and Senior Advisor to the Academic
Vice President at St. Olaf College in Northfield, Minnesota. He is the editor
or author of many books on mathematics and education, the most recent being Why
Numbers Count: Quantitative Literacy for Tomorrow's America(1997). Earlier volumes include On
the Shoulders of Giants: New Approaches to Numeracy(1991), Everybody
Counts(1989), and Calculus
for a New Century(1988).

Steen
is a member of the Mathematics Advisory Committee of The College Board.
Previously, he served as Executive Director of the Mathematical Sciences
Education Board (1992-1995), President of the Mathematical Association of
America (1985-1986), and Chair of the Council of Scientific Society Presidents
(1989). Steen received his Ph.D. in mathematics from the Massachusetts
Institute of Technology.

Forman
and Steen are coauthors of several publications dealing with the mathematical
needs of the technical workforce, including "Mathematics for Work" (Bulletin
of the International Commission on Mathematical Instruction,December 1994); "How School Mathematics Can Prepare Students for Work, Not Just
for College" (Harvard
Education Letter,May/June 1995); "Mathematics for Work and Life'' (Seventy-Five
Years of Progress: Prospects for School Mathematics,NCTM, 1995); and "Applied Academics: Myths and Realities" (Education
Week,October 19, 1996).